The Relationships between Real Estate Price and Expected Financial Asset Risk and Return: Theory and Empirical Evidence

Abstract

In pricing real estate with indifference pricing approach, market incompleteness is shown to significantly alter the conventional pricing relationships between real estate and financial asset. Specifically, we focus on the pricing implication of market comovement because comovement tends to be stronger in financial crisis when investors are especially sensitive to price declines. We find that real estate price increases with expected financial asset return but only in weak market comovement (i.e., a normal market environment) when investors enjoy diversification benefit. When market comovement is strong, real estate price strictly declines with expected financial asset return. More importantly, contrary to the conventional positive relationship from real option studies, real estate price generally declines with expected financial asset risk. With realistic market parameters, we show that there is a nonlinear relationship between real estate price and financial risk. When the market comovement is strong, real estate price only increases with financial asset risk when the risk is low but eventually declines with the risk when it becomes high. Our cross-country empirical results also show that the relationship between financial market risk and real estate price is non-monotonic, conditional on the degree of market comovement.

Appendix A

A.1. Proof of Proposition 1

Suppose that agent A keeps the land and maximize his utility by choosing the optimal investment strategy β given the optimal construction size q ^*. Differentiating Eq. (5) with regard to β gives

$$ {{\beta }^{*}} = \frac{1}{{{{\gamma }_a}}}{{\left( {S_T^u - S_T^d} \right)}^{{ - 1}}}\ln \left[ {\frac{{p_S^u\left( {S_T^u - R{{S}_0}} \right)\exp \left( {{{\gamma }_a}\left( {{{q}^{*}}Y_T^u - C\left( {{{q}^{*}}} \right)} \right)} \right)}}{{p_S^d\left( {R{{S}_0} - S_T^d} \right)\exp \left( {{{\gamma }_a}\left( {{{q}^{*}}Y_T^d - C\left( {{{q}^{*}}} \right)} \right)} \right)}}} \right] $$

(A1a)

where R is one plus the risk-less interest rate over the one period, and${{{S_T^u}} \left/ {{{{S}_0} > R > }} \right.}{{{S_T^d}} \left/ {{{{S}_0}}} \right.}$ and ${{{Y_T^u}} \left/ {{{{Y}_0} > R > }} \right.}{{{Y_T^d}} \left/ {{{{Y}_0}}} \right.}$ hold in order to eliminate arbitrage opportunities. Substituting this result back into Eq. (5) yields the following optimal utility level for the investor

$$ \matrix{ {V_a^{*}\left( {{{w}_a},k} \right) = - \exp \left[ { - {{\gamma }_a}R\left( {{{w}_a}} \right)} \right]} \\ { \times \left[ {\frac{{{{{\left( {p_S^u} \right)}}^{\pi }}\exp {{{\left( { - {{\gamma }_a}\left( {{{q}^{*}}Y_T^u - C\left( {{{q}^{*}}} \right)} \right)} \right)}}^{\pi }}{{{\left( {p_S^d} \right)}}^{{1 - \pi }}}\exp {{{\left( { - {{\gamma }_a}\left( {{{q}^{*}}Y_T^d - C\left( {{{q}^{*}}} \right)} \right)} \right)}}^{{1 - \pi }}}}}{{{{\pi }^{\pi }}{{{\left( {1 - \pi } \right)}}^{{1 - \pi }}}}}} \right]} \\ }<!end array> . $$

(A1b)

Alternatively, agent A can sell the land for kP _a , and invest all his wealth w _a  + kP _a into x units of the risk-free asset and β units of the risky asset. Differentiating Eq. (6) with regard to β, we derive

$$ {{\beta }^{{**}}} = \frac{1}{{{{\gamma }_a}}}{{\left( {S_T^u - S_T^d} \right)}^{{ - 1}}}\ln \left[ {\frac{{p_S^u\left( {S_T^u - R{{S}_0}} \right)}}{{p_S^d\left( {R{{S}_0} - S_T^d} \right)}}} \right]. $$

(A1c)

Substituting Eq. (A1c) into Eq. (6) produces the optimal utility level

$$ {{V}^{*}}\left( {{{w}_a} + k{{P}_a}} \right) = - \exp \left[ { - {{\gamma }_a}R\left( {{{w}_a} + k{{P}_a}} \right)} \right]\left[ {\frac{{{{{\left( {p_S^u} \right)}}^{\pi }}{{{\left( {p_S^d} \right)}}^{{1 - \pi }}}}}{{{{\pi }^{\pi }}{{{\left( {1 - \pi } \right)}}^{{1 - \pi }}}}}} \right]. $$

(A1d)

According to the definition of indifference ask prices, we have

$$ V_a^{*}\left( {{{w}_a},k} \right) = {{V}^{*}}\left( {{{w}_a} + k{{P}_a}} \right). $$

(A1e)

Solving this equation, we can obtain the indifference ask price as follows

$$ \matrix{ {{{P}_a} \approx - \frac{1}{{{{\gamma }_a}Rk}}\left[ {\pi \cdot \ln \left[ {\exp \left( { - {{\gamma }_a}\left( {{{q}^{*}}Y_T^u - C\left( {{{q}^{*}}} \right)} \right)} \right)} \right] + \left( {1 - \pi } \right) \cdot \ln \left[ {\exp \left( { - {{\gamma }_a}\left( {{{q}^{*}}Y_T^d - C\left( {{{q}^{*}}} \right)} \right)} \right)} \right]} \right]} \\ { = \frac{1}{{Rk}}\left[ {{{{\text E}}_{\pi }}\left( {{{q}^{*}}{{{\tilde{Y}}}_T} - C\left( {{{q}^{*}}} \right)} \right)} \right].} \\ }<!end array> $$

(A1f)

On the other hand, agent B is indifferent between foregoing the vacant land at time 0 and paying kP _b for the land at time T; and thus similarly, we can obtain the bid price for the land as

$$ \matrix{ {{{P}_b} \approx - \frac{1}{{{{\gamma }_b}Rk}}\left[ {\pi \cdot \ln \left[ {\exp \left( { - {{\gamma }_b}\left( {{{q}^{*}}Y_T^u - C\left( {{{q}^{*}}} \right)} \right)} \right)} \right] + \left( {1 - \pi } \right) \cdot \ln \left[ {\exp \left( { - {{\gamma }_b}\left( {{{q}^{*}}Y_T^d - C\left( {{{q}^{*}}} \right)} \right)} \right)} \right]} \right]} \\ { = \frac{1}{{Rk}}\left[ {{{{\text E}}_{\pi }}\left( {{{q}^{*}}{{{\tilde{Y}}}_T} - C\left( {{{q}^{*}}} \right)} \right)} \right].} \\ }<!end array> $$

(A1g)

Q.E.D.

A.2. Proof of Proposition 3

The mathematical implications for the Proposition 3 are represented in the following comparative statics results. We find that, $\frac{{\partial {{P}_a}}}{{\partial S_T^u}} = \frac{{\partial {{P}_b}}}{{\partial S_T^u}} < 0$, $\frac{{\partial {{P}_a}}}{{\partial S_T^d}} = \frac{{\partial {{P}_b}}}{{\partial S_T^d}} < 0$, and $\frac{{\partial {{P}_a}}}{{\partial \eta }} = \frac{{\partial {{P}_b}}}{{\partial \eta }} > 0$.

Differentiating Eq. (9) produces the following derivatives

$$ \frac{{\partial {{P}_a}}}{{\partial S_T^u}} = \frac{{\partial {{P}_b}}}{{\partial S_T^u}} = - \frac{{\left( {R{{S}_0} - S_T^d} \right)\left( {{{q}^{*}}Y_T^u - {{q}^{*}}Y_T^d} \right)}}{{Rk{{{\left( {S_T^u - S_T^d} \right)}}^2}}} < 0 $$

(A2a)

$$ \frac{{\partial {{P}_a}}}{{\partial S_T^d}} = \frac{{\partial {{P}_b}}}{{\partial S_T^d}} = - \frac{{\left( {S_T^u - R{{S}_0}} \right)\left( {{{q}^{*}}Y_T^u - {{q}^{*}}Y_T^d} \right)}}{{Rk{{{\left( {S_T^u - S_T^d} \right)}}^2}}} < 0 $$

(A2b)

We use η to represent the increase of the risk of the risky financial asset. If we increase $S_T^u$ by η, to keep the expected value of ${{\tilde{S}}_T}$ the same, we need to reduce the payoff $S_T^d$ by${{p}^u}\eta /{{p}^d}$.

$$ \frac{{d{{P}_a}}}{{d\eta }} = \frac{{d{{P}_b}}}{{d\eta }} = \frac{1}{{Rk}}\left[ {\frac{{d\pi }}{{d\eta }}\left( {{{q}^{*}}Y_T^u - C\left( {{{q}^{*}}} \right)} \right) - \frac{{d\pi }}{{d\eta }}\left( {{{q}^{*}}Y_T^d - C\left( {{{q}^{*}}} \right)} \right)} \right] = \frac{{d\pi }}{{d\eta }}\left( {{{q}^{*}}Y_T^u - {{q}^{*}}Y_T^d} \right) > 0 $$

(A2c)

$$ {\text{where }}\frac{{d\pi }}{{d\eta }} = \frac{{\left[ {\frac{{p_s^u}}{{p_s^d}}\left( {S_T^u - R{{S}_0}} \right) - \left( {R{{S}_0} - S_T^d} \right)} \right]}}{{{{{\left[ {S_T^u - S_T^d + \eta \left( {1 + \frac{{p_s^u}}{{p_s^d}}} \right)} \right]}}^2}}} = \frac{{\left( {p_s^uS_T^u + p_s^dS_T^d - R{{S}_0}} \right)}}{{p_s^d{{{\left[ {S_T^u - S_T^d + \eta \left( {1 + \frac{{p_s^u}}{{p_s^d}}} \right)} \right]}}^2}}} > 0, $$

since $p_s^uS_T^u + p_s^dS_T^d > R{{S}_0}$ by assumption that stock is riskier than riskless asset.

Q.E.D

A.3. Proof of Proposition 4

The prices of the land in the incomplete market setting with a risky hedging financial asset are expressed in the following equations.

$$ \begin{array}{*{20}{c}} {{{P}_{i}} = - {{{\text{E}}}_{Q}}\left[ {\frac{1}{{{{\gamma }_{a}}Rk}}\ln {{{\text{E}}}_{Q}}\left[ {\exp \left( { - {{\gamma }_{i}}\left( {{{q}^{*}}{{{\widetilde{Y}}}_{T}} - C\left( {{{q}^{*}}} \right)} \right)} \right)\left| {{{{\widetilde{S}}}_{T}}} \right.} \right]} \right]} \\ { \approx \frac{1}{{Rk}}\left[ {{{{\text{E}}}_{Q}}\left( {{{q}^{*}}{{{\widetilde{Y}}}_{T}} - C\left( {{{q}^{*}}} \right)\left| {{{{\widetilde{S}}}_{T}}} \right.} \right) - \frac{{{{\gamma }_{i}}}}{2}{{{\text{E}}}_{Q}}\left[ {Va{{r}_{Q}}\left( {{{q}^{*}}{{{\widetilde{Y}}}_{T}} - C\left( {{{q}^{*}}} \right)\left| {{{{\widetilde{S}}}_{T}}} \right.} \right)} \right]} \right]} \\ { = \frac{1}{{Rk}}\left[ {\left( {{{Q}_{1}} + {{Q}_{2}}} \right)\left( {{{q}^{*}}Y_{T}^{u} - C\left( {{{q}^{*}}} \right)} \right) + \left( {{{Q}_{3}} + {{Q}_{4}}} \right)\left( {{{q}^{*}}Y_{T}^{d} - C\left( {{{q}^{*}}} \right)} \right) - \frac{{{{\gamma }_{i}}}}{2}\left( {\frac{{{{Q}_{1}}{{Q}_{2}}}}{\pi } + \frac{{{{Q}_{3}}{{Q}_{4}}}}{{1 - \pi }}} \right){{{\left( {{{q}^{*}}\left( {Y_{T}^{u} - Y_{T}^{d}} \right)} \right)}}^{2}}} \right].} \\ { = \frac{1}{{Rk}}\left[ {\left( {\pi {{p}^{{u\left| u \right.}}} + \left( {1 - \pi } \right){{p}^{{u\left| d \right.}}}} \right)\left( {{{q}^{*}}Y_{T}^{u} - C\left( {{{q}^{*}}} \right)} \right) + \left( {\pi {{p}^{{d\left| u \right.}}} + \left( {1 - \pi } \right){{p}^{{d\left| d \right.}}}} \right)\left( {{{q}^{*}}Y_{T}^{d} - C\left( {{{q}^{*}}} \right)} \right)} \right.} \\ {\left. { - \frac{{{{\gamma }_{i}}}}{2}\left( {\pi \cdot {{p}^{{u\left| u \right.}}}{{p}^{{d\left| u \right.}}} + \left( {1 - \pi } \right)\cdot {{p}^{{u\left| d \right.}}}{{p}^{{d\left| d \right.}}}} \right){{{\left( {{{q}^{*}}Y_{T}^{u} - {{q}^{*}}Y_{T}^{d}} \right)}}^{2}}} \right]} \\ \end{array} $$

The mathematical implications for the Proposition 4 are represented in the following comparative statics results after differentiating the above pricing equations.

$$ \frac{{\partial {{P}_a}}}{{\partial {{\gamma }_a}}} = \frac{{\partial {{P}_b}}}{{\partial {{\gamma }_b}}} < 0,\frac{{d{{P}_a}}}{{dz}} < 0,\frac{{d{{P}_b}}}{{dz}} < 0,\frac{{\partial {{P}_a}}}{{\partial S_T^u}}\frac{ > }{ < }0,\frac{{\partial {{P}_b}}}{{\partial S_T^u}}\frac{ > }{ < }0,\frac{{\partial {{P}_a}}}{{\partial S_T^d}}\frac{ > }{ < }0,\frac{{\partial {{P}_b}}}{{\partial S_T^d}}\frac{ > }{ < }0,\frac{{d{{P}_i}}}{{d\eta }}\frac{ > }{ < }0. $$

$$ \frac{{\partial {{P}_a}}}{{\partial {{\gamma }_a}}} = - \frac{1}{{2Rk}}{{{\text E}}_Q}\left[ {Va{{r}_Q}\left( {{{q}^{*}}{{{\tilde{Y}}}_T} - C\left( {{{q}^{*}}} \right)\left| {{{{\tilde{S}}}_T}} \right.} \right)} \right] < 0 $$

(A3a)

$$ \frac{{\partial {{P}_b}}}{{\partial {{\gamma }_b}}} = - \frac{1}{{2Rk}}{{{\text E}}_Q}\left[ {Va{{r}_Q}\left( {{{q}^{*}}{{{\tilde{Y}}}_T} - C\left( {{{q}^{*}}} \right)\left| {{{{\tilde{S}}}_T}} \right.} \right)} \right] < 0 $$

(A3b)

In order to change the volatility while keeping the expected mean of the random variable ${{\tilde{Y}}_T}$ unchanged, we introduce the variable z to represent the increase in $Y_T^u$, whereas decrease $Y_T^d$ by ${{{z\left( {\pi {{p}^{{u\left| u \right.}}} + \left( {1 - \pi } \right){{p}^{{u\left| d \right.}}}} \right)}} \left/ {{\left( {\pi {{p}^{{d\left| u \right.}}} + \left( {1 - \pi } \right){{p}^{{d\left| d \right.}}}} \right)}} \right.}$ at the same time. Hence, z represents the change in the nondiversifiable idiosyncratic risk of the real asset. Substituting these changes into Eq. (12) and then differentiating the resultant equation, we obtain the following results.

Let ${{h}^u} = \pi {{p}^{{u\left| u \right.}}} + \left( {1 - \pi } \right){{p}^{{u\left| d \right.}}}$ and ${{h}^d} = \pi {{p}^{{d\left| u \right.}}} + \left( {1 - \pi } \right){{p}^{{d\left| d \right.}}}$. Then the following derivatives can be produced

$$ \begin{array}{*{20}{c}} {\frac{{d{{P}_{i}}}}{{dz}} = - \frac{{{{\gamma }_{i}}\pi {{p}^{{u\left| u \right.}}}{{p}^{{d\left| u \right.}}}}}{{Rk}}\left[ {\left( {{{q}^{*}}Y_{T}^{u} - {{q}^{*}}Y_{T}^{d}} \right) + {{q}^{*}}z\left( {1 + \frac{{{{h}^{u}}}}{{{{h}^{d}}}}} \right)} \right]\left( {{{q}^{*}} + \frac{{{{q}^{*}}{{h}^{u}}}}{{{{h}^{d}}}}} \right) - } \\ {\frac{{{{\gamma }_{i}}\left( {1 - \pi } \right){{p}^{{u\left| d \right.}}}{{p}^{{d\left| d \right.}}}}}{{Rk}}\left[ {\left( {{{q}^{*}}Y_{T}^{u} - {{q}^{*}}Y_{T}^{d}} \right) + {{q}^{*}}z\left( {1 + \frac{{{{h}^{u}}}}{{{{h}^{d}}}}} \right)} \right]\left( {{{q}^{*}} + \frac{{{{q}^{*}}{{h}^{u}}}}{{{{h}^{d}}}}} \right) < 0} \\ \end{array} $$

(A3c)

$$ \frac{{\partial {{P}_i}}}{{\partial S_T^u}} = - \frac{{{{P}_i}}}{{\left( {S_T^u - S_T^d} \right)}} + \frac{{{{p}^{{u\left| d \right.}}}\left( {q^{*}Y_T^u - C\left( {q^{*}} \right)} \right) + {{p}^{{d\left| d \right.}}}\left( {q^{*}Y_T^d - C\left( {q^{*}} \right)} \right)}}{{Rk\left( {S_T^u - S_T^d} \right)}} - \frac{{{{\gamma }_i}{{p}^{{u\left| d \right.}}}{{p}^{{d\left| d \right.}}}{{{\left( {q^{*}Y_T^u - q^{*}Y_T^d} \right)}}^2}}}{{2Rk\left( {S_T^u - S_T^d} \right)}}\frac{ > }{ < }0 $$

(A3d)

$$ \frac{{\partial {{P}_i}}}{{\partial S_T^d}} = \frac{{{{P}_i}}}{{\left( {S_T^u - S_T^d} \right)}} - \frac{{{{p}^{{u\left| u \right.}}}\left( {{{q}^{*}}Y_T^u - C\left( {{{q}^{*}}} \right)} \right) + {{p}^{{d\left| u \right.}}}\left( {{{q}^{*}}Y_T^d - C\left( {{{q}^{*}}} \right)} \right)}}{{Rk\left( {S_T^u - S_T^d} \right)}} + \frac{{{{\gamma }_i}{{p}^{{u\left| u \right.}}}{{p}^{{d\left| u \right.}}}{{{\left( {{{q}^{*}}Y_T^u - {{q}^{*}}Y_T^d} \right)}}^2}}}{{2Rk\left( {S_T^u - S_T^d} \right)}}\frac{ > }{ < }0 $$

(A3e)

We use η to represent the increase of the risk of the risky financial asset. If we increase $S_T^u$ by η, to keep the expected value of ${{\tilde{S}}_T}$ the same, we need to reduce the payoff $S_T^d$ by ${{p}^u}\eta /{{p}^d}$. The derivative is shown as the followings:

$$ \matrix{ {\frac{{d{{P}_i}}}{{d\eta }} = \frac{1}{{Rk}}\left[ {\left( {\frac{{d\pi }}{{d\eta }}{{p}^{{u\left| u \right.}}} - \frac{{d\pi }}{{d\eta }}{{p}^{{u\left| d \right.}}}} \right)\left( {{{q}^{*}}Y_T^u - C\left( {{{q}^{*}}} \right)} \right) + \left( {\frac{{d\pi }}{{d\eta }}{{p}^{{d\left| u \right.}}} - \frac{{d\pi }}{{d\eta }}{{p}^{{d\left| d \right.}}}} \right)\left( {{{q}^{*}}Y_T^d - C\left( {{{q}^{*}}} \right)} \right)} \right.} \hfill \\ {\left. { - \frac{{{{\gamma }_i}}}{2}\left( {\frac{{d\pi }}{{d\eta }}{{p}^{{u\left| u \right.}}}{{p}^{{d\left| u \right.}}} - \frac{{d\pi }}{{d\eta }}{{p}^{{u\left| d \right.}}}{{p}^{{d\left| d \right.}}}} \right){{{\left( {{{q}^{*}}Y_T^u - {{q}^{*}}Y_T^d} \right)}}^2}} \right]\frac{ > }{ < }0.} \hfill \\ }<!end array> $$

(A3f)

We have shown the exact form of $\frac{{d\pi }}{{d\eta }}$ in appendix A.2.

Q.E.D

A.4. Proof of the Relationship between joint Probabilities and Covariance Structure of Real Estate and Financial Asset

According to the definition of covariance, we have the covariance of ${{\tilde{Y}}_T}$ and ${{\tilde{S}}_T}$as follows

$$ \begin{array}{*{20}{c}} {Cov = {{p}^{{uu}}}\left( {Y_{T}^{u} - E{{{\widetilde{Y}}}_{T}}} \right)\left( {S_{T}^{u} - E{{{\widetilde{S}}}_{T}}} \right) + {{p}^{{ud}}}\left( {Y_{T}^{u} - E{{{\widetilde{Y}}}_{T}}} \right)\left( {S_{T}^{d} - E{{{\widetilde{S}}}_{T}}} \right)} \\ { + {{p}^{{du}}}\left( {Y_{T}^{d} - E{{{\widetilde{Y}}}_{T}}} \right)\left( {S_{T}^{u} - E{{{\widetilde{S}}}_{T}}} \right) + {{p}^{{dd}}}\left( {Y_{T}^{d} - E{{{\widetilde{Y}}}_{T}}} \right)\left( {S_{T}^{d} - E{{{\widetilde{S}}}_{T}}} \right),} \\ \end{array} $$

(A4a)

where ${{p}^{{uu}}},{ }{{p}^{{ud}}},{ }{{p}^{{du}}},{\text{ and }}{{p}^{{dd}}}$represent the joint probabilities of the outcomes $\left( {Y_T^u,S_T^u} \right),\left( {Y_T^u,S_T^d} \right),\left( {Y_T^d,S_T^u} \right),{\text{ and }}\left( {Y_T^d,S_T^d} \right).$ If these two variables have a high positive correlation (i.e., strong market comovement) that implies small values of p ^ud and p ^du, we can approximate the above equation as

$$ Cov \approx {{p}^{{uu}}}\left( {Y_T^u - E{{{\tilde{Y}}}_T}} \right)\left( {S_T^u - E{{{\tilde{S}}}_T}} \right) + {{p}^{{dd}}}\left( {Y_T^d - E{{{\tilde{Y}}}_T}} \right)\left( {S_T^d - E{{{\tilde{S}}}_T}} \right). $$

(A4b)

Given that the up and down probabilities of ${{\tilde{Y}}_T}{\text{ and }}{{\tilde{S}}_T}$ remain unchanged and that their increasing and decreasing proportions are also unchanged, this suggests that an increasing joint up or down probability of these two variables leads to a greater positive correlation between these two variables. If these two variables tend to be negatively correlated so that ${{p}^{{uu}}}$ and ${{p}^{{dd}}}$are very low, this equation can be approximately expressed as follows

$$ Cov \approx {{p}^{{ud}}}\left( {Y_T^u - E{{{\tilde{Y}}}_T}} \right)\left( {S_T^d - E{{{\tilde{S}}}_T}} \right) + {{p}^{{du}}}\left( {Y_T^d - E{{{\tilde{Y}}}_T}} \right)\left( {S_T^u - E{{{\tilde{S}}}_T}} \right). $$

(A4c)

Given those conditions remaining unchanged, this suggests that an increase in ${{p}^{{ud}}}$ and ${{p}^{{du}}}$results in a greater absolute-value correlation between these two variables. It is easy to find that our results can be also applied to examine the relationship between the conditional probabilities and conariance structure.